Integrand size = 343, antiderivative size = 31 \[ \int \frac {-3 x \log (3)+e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} \left (6 x^2-6 x \log (3)\right )+\left (6 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} \log (3)+6 x \log (3)\right ) \log (x)}{36 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)+36 x^2 \log (3)+\left (-12 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)-12 x^2 \log (3)\right ) \log ^2(x)+\left (e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)+x^2 \log (3)\right ) \log ^4(x)+\left (12 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)+12 x^2 \log (3)+\left (-2 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)-2 x^2 \log (3)\right ) \log ^2(x)\right ) \log \left (e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}}+x\right )+\left (e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)+x^2 \log (3)\right ) \log ^2\left (e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}}+x\right )} \, dx=\frac {3}{6-\log ^2(x)+\log \left (e^{-2+2 x-\frac {x^2}{\log (3)}}+x\right )} \] Output:
3/(ln(x+exp(2*x-2-x^2/ln(3)))-ln(x)^2+6)
Time = 0.16 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {-3 x \log (3)+e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} \left (6 x^2-6 x \log (3)\right )+\left (6 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} \log (3)+6 x \log (3)\right ) \log (x)}{36 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)+36 x^2 \log (3)+\left (-12 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)-12 x^2 \log (3)\right ) \log ^2(x)+\left (e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)+x^2 \log (3)\right ) \log ^4(x)+\left (12 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)+12 x^2 \log (3)+\left (-2 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)-2 x^2 \log (3)\right ) \log ^2(x)\right ) \log \left (e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}}+x\right )+\left (e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)+x^2 \log (3)\right ) \log ^2\left (e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}}+x\right )} \, dx=-\frac {3}{-6+\log ^2(x)-\log \left (e^{-2+2 x-\frac {x^2}{\log (3)}}+x\right )} \] Input:
Integrate[(-3*x*Log[3] + E^((-x^2 + (-2 + 2*x)*Log[3])/Log[3])*(6*x^2 - 6* x*Log[3]) + (6*E^((-x^2 + (-2 + 2*x)*Log[3])/Log[3])*Log[3] + 6*x*Log[3])* Log[x])/(36*E^((-x^2 + (-2 + 2*x)*Log[3])/Log[3])*x*Log[3] + 36*x^2*Log[3] + (-12*E^((-x^2 + (-2 + 2*x)*Log[3])/Log[3])*x*Log[3] - 12*x^2*Log[3])*Lo g[x]^2 + (E^((-x^2 + (-2 + 2*x)*Log[3])/Log[3])*x*Log[3] + x^2*Log[3])*Log [x]^4 + (12*E^((-x^2 + (-2 + 2*x)*Log[3])/Log[3])*x*Log[3] + 12*x^2*Log[3] + (-2*E^((-x^2 + (-2 + 2*x)*Log[3])/Log[3])*x*Log[3] - 2*x^2*Log[3])*Log[ x]^2)*Log[E^((-x^2 + (-2 + 2*x)*Log[3])/Log[3]) + x] + (E^((-x^2 + (-2 + 2 *x)*Log[3])/Log[3])*x*Log[3] + x^2*Log[3])*Log[E^((-x^2 + (-2 + 2*x)*Log[3 ])/Log[3]) + x]^2),x]
Output:
-3/(-6 + Log[x]^2 - Log[E^(-2 + 2*x - x^2/Log[3]) + x])
Time = 1.62 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {7239, 27, 27, 7237}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {e^{\frac {(2 x-2) \log (3)-x^2}{\log (3)}} \left (6 x^2-6 x \log (3)\right )+\left (6 \log (3) e^{\frac {(2 x-2) \log (3)-x^2}{\log (3)}}+6 x \log (3)\right ) \log (x)-3 x \log (3)}{\left (x^2 \log (3)+x \log (3) e^{\frac {(2 x-2) \log (3)-x^2}{\log (3)}}\right ) \log ^4(x)+\left (-12 x^2 \log (3)-12 x \log (3) e^{\frac {(2 x-2) \log (3)-x^2}{\log (3)}}\right ) \log ^2(x)+\left (x^2 \log (3)+x \log (3) e^{\frac {(2 x-2) \log (3)-x^2}{\log (3)}}\right ) \log ^2\left (e^{\frac {(2 x-2) \log (3)-x^2}{\log (3)}}+x\right )+\left (\left (-2 x^2 \log (3)-2 x \log (3) e^{\frac {(2 x-2) \log (3)-x^2}{\log (3)}}\right ) \log ^2(x)+12 x^2 \log (3)+12 x \log (3) e^{\frac {(2 x-2) \log (3)-x^2}{\log (3)}}\right ) \log \left (e^{\frac {(2 x-2) \log (3)-x^2}{\log (3)}}+x\right )+36 x^2 \log (3)+36 x \log (3) e^{\frac {(2 x-2) \log (3)-x^2}{\log (3)}}} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-3 x \log (3) e^{\frac {x^2}{\log (3)}+2}+6 \log (3) \left (x e^{\frac {x^2}{\log (3)}+2}+e^{2 x}\right ) \log (x)+6 e^{2 x} x (x-\log (3))}{x \log (3) \left (x e^{\frac {x^2}{\log (3)}+2}+e^{2 x}\right ) \left (\log \left (e^{-\frac {x^2}{\log (3)}+2 x-2}+x\right )-\log ^2(x)+6\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {3 \left (2 e^{2 x} (x-\log (3)) x-e^{\frac {x^2}{\log (3)}+2} \log (3) x+2 \left (e^{\frac {x^2}{\log (3)}+2} x+e^{2 x}\right ) \log (3) \log (x)\right )}{x \left (e^{\frac {x^2}{\log (3)}+2} x+e^{2 x}\right ) \left (-\log ^2(x)+\log \left (x+e^{-\frac {x^2}{\log (3)}+2 x-2}\right )+6\right )^2}dx}{\log (3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \int \frac {2 e^{2 x} (x-\log (3)) x-e^{\frac {x^2}{\log (3)}+2} \log (3) x+2 \left (e^{\frac {x^2}{\log (3)}+2} x+e^{2 x}\right ) \log (3) \log (x)}{x \left (e^{\frac {x^2}{\log (3)}+2} x+e^{2 x}\right ) \left (-\log ^2(x)+\log \left (x+e^{-\frac {x^2}{\log (3)}+2 x-2}\right )+6\right )^2}dx}{\log (3)}\) |
| \(\Big \downarrow \) 7237 |
| \(\displaystyle \frac {3}{\log \left (e^{-\frac {x^2}{\log (3)}+2 x-2}+x\right )-\log ^2(x)+6}\) |
Input:
Int[(-3*x*Log[3] + E^((-x^2 + (-2 + 2*x)*Log[3])/Log[3])*(6*x^2 - 6*x*Log[ 3]) + (6*E^((-x^2 + (-2 + 2*x)*Log[3])/Log[3])*Log[3] + 6*x*Log[3])*Log[x] )/(36*E^((-x^2 + (-2 + 2*x)*Log[3])/Log[3])*x*Log[3] + 36*x^2*Log[3] + (-1 2*E^((-x^2 + (-2 + 2*x)*Log[3])/Log[3])*x*Log[3] - 12*x^2*Log[3])*Log[x]^2 + (E^((-x^2 + (-2 + 2*x)*Log[3])/Log[3])*x*Log[3] + x^2*Log[3])*Log[x]^4 + (12*E^((-x^2 + (-2 + 2*x)*Log[3])/Log[3])*x*Log[3] + 12*x^2*Log[3] + (-2 *E^((-x^2 + (-2 + 2*x)*Log[3])/Log[3])*x*Log[3] - 2*x^2*Log[3])*Log[x]^2)* Log[E^((-x^2 + (-2 + 2*x)*Log[3])/Log[3]) + x] + (E^((-x^2 + (-2 + 2*x)*Lo g[3])/Log[3])*x*Log[3] + x^2*Log[3])*Log[E^((-x^2 + (-2 + 2*x)*Log[3])/Log [3]) + x]^2),x]
Output:
3/(6 - Log[x]^2 + Log[E^(-2 + 2*x - x^2/Log[3]) + x])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 24.41 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16
| method | result | size |
| parallelrisch | \(-\frac {3}{\ln \left (x \right )^{2}-\ln \left ({\mathrm e}^{\frac {\left (-2+2 x \right ) \ln \left (3\right )-x^{2}}{\ln \left (3\right )}}+x \right )-6}\) | \(36\) |
| risch | \(-\frac {3}{\ln \left (x \right )^{2}-\ln \left ({\mathrm e}^{\frac {2 x \ln \left (3\right )-x^{2}-2 \ln \left (3\right )}{\ln \left (3\right )}}+x \right )-6}\) | \(37\) |
Input:
int(((6*ln(3)*exp(((-2+2*x)*ln(3)-x^2)/ln(3))+6*x*ln(3))*ln(x)+(-6*x*ln(3) +6*x^2)*exp(((-2+2*x)*ln(3)-x^2)/ln(3))-3*x*ln(3))/((x*ln(3)*exp(((-2+2*x) *ln(3)-x^2)/ln(3))+x^2*ln(3))*ln(exp(((-2+2*x)*ln(3)-x^2)/ln(3))+x)^2+((-2 *x*ln(3)*exp(((-2+2*x)*ln(3)-x^2)/ln(3))-2*x^2*ln(3))*ln(x)^2+12*x*ln(3)*e xp(((-2+2*x)*ln(3)-x^2)/ln(3))+12*x^2*ln(3))*ln(exp(((-2+2*x)*ln(3)-x^2)/l n(3))+x)+(x*ln(3)*exp(((-2+2*x)*ln(3)-x^2)/ln(3))+x^2*ln(3))*ln(x)^4+(-12* x*ln(3)*exp(((-2+2*x)*ln(3)-x^2)/ln(3))-12*x^2*ln(3))*ln(x)^2+36*x*ln(3)*e xp(((-2+2*x)*ln(3)-x^2)/ln(3))+36*x^2*ln(3)),x,method=_RETURNVERBOSE)
Output:
-3/(ln(x)^2-ln(exp(((-2+2*x)*ln(3)-x^2)/ln(3))+x)-6)
Time = 0.08 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {-3 x \log (3)+e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} \left (6 x^2-6 x \log (3)\right )+\left (6 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} \log (3)+6 x \log (3)\right ) \log (x)}{36 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)+36 x^2 \log (3)+\left (-12 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)-12 x^2 \log (3)\right ) \log ^2(x)+\left (e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)+x^2 \log (3)\right ) \log ^4(x)+\left (12 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)+12 x^2 \log (3)+\left (-2 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)-2 x^2 \log (3)\right ) \log ^2(x)\right ) \log \left (e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}}+x\right )+\left (e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)+x^2 \log (3)\right ) \log ^2\left (e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}}+x\right )} \, dx=-\frac {3}{\log \left (x\right )^{2} - \log \left (x + e^{\left (-\frac {x^{2} - 2 \, {\left (x - 1\right )} \log \left (3\right )}{\log \left (3\right )}\right )}\right ) - 6} \] Input:
integrate(((6*log(3)*exp(((2*x-2)*log(3)-x^2)/log(3))+6*x*log(3))*log(x)+( -6*x*log(3)+6*x^2)*exp(((2*x-2)*log(3)-x^2)/log(3))-3*x*log(3))/((x*log(3) *exp(((2*x-2)*log(3)-x^2)/log(3))+x^2*log(3))*log(exp(((2*x-2)*log(3)-x^2) /log(3))+x)^2+((-2*x*log(3)*exp(((2*x-2)*log(3)-x^2)/log(3))-2*x^2*log(3)) *log(x)^2+12*x*log(3)*exp(((2*x-2)*log(3)-x^2)/log(3))+12*x^2*log(3))*log( exp(((2*x-2)*log(3)-x^2)/log(3))+x)+(x*log(3)*exp(((2*x-2)*log(3)-x^2)/log (3))+x^2*log(3))*log(x)^4+(-12*x*log(3)*exp(((2*x-2)*log(3)-x^2)/log(3))-1 2*x^2*log(3))*log(x)^2+36*x*log(3)*exp(((2*x-2)*log(3)-x^2)/log(3))+36*x^2 *log(3)),x, algorithm="fricas")
Output:
-3/(log(x)^2 - log(x + e^(-(x^2 - 2*(x - 1)*log(3))/log(3))) - 6)
Time = 0.48 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {-3 x \log (3)+e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} \left (6 x^2-6 x \log (3)\right )+\left (6 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} \log (3)+6 x \log (3)\right ) \log (x)}{36 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)+36 x^2 \log (3)+\left (-12 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)-12 x^2 \log (3)\right ) \log ^2(x)+\left (e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)+x^2 \log (3)\right ) \log ^4(x)+\left (12 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)+12 x^2 \log (3)+\left (-2 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)-2 x^2 \log (3)\right ) \log ^2(x)\right ) \log \left (e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}}+x\right )+\left (e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)+x^2 \log (3)\right ) \log ^2\left (e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}}+x\right )} \, dx=\frac {3}{- \log {\left (x \right )}^{2} + \log {\left (x + e^{\frac {- x^{2} + \left (2 x - 2\right ) \log {\left (3 \right )}}{\log {\left (3 \right )}}} \right )} + 6} \] Input:
integrate(((6*ln(3)*exp(((2*x-2)*ln(3)-x**2)/ln(3))+6*x*ln(3))*ln(x)+(-6*x *ln(3)+6*x**2)*exp(((2*x-2)*ln(3)-x**2)/ln(3))-3*x*ln(3))/((x*ln(3)*exp((( 2*x-2)*ln(3)-x**2)/ln(3))+x**2*ln(3))*ln(exp(((2*x-2)*ln(3)-x**2)/ln(3))+x )**2+((-2*x*ln(3)*exp(((2*x-2)*ln(3)-x**2)/ln(3))-2*x**2*ln(3))*ln(x)**2+1 2*x*ln(3)*exp(((2*x-2)*ln(3)-x**2)/ln(3))+12*x**2*ln(3))*ln(exp(((2*x-2)*l n(3)-x**2)/ln(3))+x)+(x*ln(3)*exp(((2*x-2)*ln(3)-x**2)/ln(3))+x**2*ln(3))* ln(x)**4+(-12*x*ln(3)*exp(((2*x-2)*ln(3)-x**2)/ln(3))-12*x**2*ln(3))*ln(x) **2+36*x*ln(3)*exp(((2*x-2)*ln(3)-x**2)/ln(3))+36*x**2*ln(3)),x)
Output:
3/(-log(x)**2 + log(x + exp((-x**2 + (2*x - 2)*log(3))/log(3))) + 6)
Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.42 \[ \int \frac {-3 x \log (3)+e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} \left (6 x^2-6 x \log (3)\right )+\left (6 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} \log (3)+6 x \log (3)\right ) \log (x)}{36 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)+36 x^2 \log (3)+\left (-12 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)-12 x^2 \log (3)\right ) \log ^2(x)+\left (e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)+x^2 \log (3)\right ) \log ^4(x)+\left (12 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)+12 x^2 \log (3)+\left (-2 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)-2 x^2 \log (3)\right ) \log ^2(x)\right ) \log \left (e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}}+x\right )+\left (e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)+x^2 \log (3)\right ) \log ^2\left (e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}}+x\right )} \, dx=-\frac {3 \, \log \left (3\right )}{\log \left (3\right ) \log \left (x\right )^{2} + x^{2} - \log \left (3\right ) \log \left (x e^{\left (\frac {x^{2}}{\log \left (3\right )} + 2\right )} + e^{\left (2 \, x\right )}\right ) - 4 \, \log \left (3\right )} \] Input:
integrate(((6*log(3)*exp(((2*x-2)*log(3)-x^2)/log(3))+6*x*log(3))*log(x)+( -6*x*log(3)+6*x^2)*exp(((2*x-2)*log(3)-x^2)/log(3))-3*x*log(3))/((x*log(3) *exp(((2*x-2)*log(3)-x^2)/log(3))+x^2*log(3))*log(exp(((2*x-2)*log(3)-x^2) /log(3))+x)^2+((-2*x*log(3)*exp(((2*x-2)*log(3)-x^2)/log(3))-2*x^2*log(3)) *log(x)^2+12*x*log(3)*exp(((2*x-2)*log(3)-x^2)/log(3))+12*x^2*log(3))*log( exp(((2*x-2)*log(3)-x^2)/log(3))+x)+(x*log(3)*exp(((2*x-2)*log(3)-x^2)/log (3))+x^2*log(3))*log(x)^4+(-12*x*log(3)*exp(((2*x-2)*log(3)-x^2)/log(3))-1 2*x^2*log(3))*log(x)^2+36*x*log(3)*exp(((2*x-2)*log(3)-x^2)/log(3))+36*x^2 *log(3)),x, algorithm="maxima")
Output:
-3*log(3)/(log(3)*log(x)^2 + x^2 - log(3)*log(x*e^(x^2/log(3) + 2) + e^(2* x)) - 4*log(3))
Time = 0.88 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {-3 x \log (3)+e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} \left (6 x^2-6 x \log (3)\right )+\left (6 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} \log (3)+6 x \log (3)\right ) \log (x)}{36 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)+36 x^2 \log (3)+\left (-12 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)-12 x^2 \log (3)\right ) \log ^2(x)+\left (e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)+x^2 \log (3)\right ) \log ^4(x)+\left (12 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)+12 x^2 \log (3)+\left (-2 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)-2 x^2 \log (3)\right ) \log ^2(x)\right ) \log \left (e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}}+x\right )+\left (e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)+x^2 \log (3)\right ) \log ^2\left (e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}}+x\right )} \, dx=-\frac {3}{\log \left (x\right )^{2} - \log \left (x e^{2} + e^{\left (-\frac {x^{2} - 2 \, x \log \left (3\right )}{\log \left (3\right )}\right )}\right ) - 4} \] Input:
integrate(((6*log(3)*exp(((2*x-2)*log(3)-x^2)/log(3))+6*x*log(3))*log(x)+( -6*x*log(3)+6*x^2)*exp(((2*x-2)*log(3)-x^2)/log(3))-3*x*log(3))/((x*log(3) *exp(((2*x-2)*log(3)-x^2)/log(3))+x^2*log(3))*log(exp(((2*x-2)*log(3)-x^2) /log(3))+x)^2+((-2*x*log(3)*exp(((2*x-2)*log(3)-x^2)/log(3))-2*x^2*log(3)) *log(x)^2+12*x*log(3)*exp(((2*x-2)*log(3)-x^2)/log(3))+12*x^2*log(3))*log( exp(((2*x-2)*log(3)-x^2)/log(3))+x)+(x*log(3)*exp(((2*x-2)*log(3)-x^2)/log (3))+x^2*log(3))*log(x)^4+(-12*x*log(3)*exp(((2*x-2)*log(3)-x^2)/log(3))-1 2*x^2*log(3))*log(x)^2+36*x*log(3)*exp(((2*x-2)*log(3)-x^2)/log(3))+36*x^2 *log(3)),x, algorithm="giac")
Output:
-3/(log(x)^2 - log(x*e^2 + e^(-(x^2 - 2*x*log(3))/log(3))) - 4)
Time = 4.45 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {-3 x \log (3)+e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} \left (6 x^2-6 x \log (3)\right )+\left (6 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} \log (3)+6 x \log (3)\right ) \log (x)}{36 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)+36 x^2 \log (3)+\left (-12 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)-12 x^2 \log (3)\right ) \log ^2(x)+\left (e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)+x^2 \log (3)\right ) \log ^4(x)+\left (12 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)+12 x^2 \log (3)+\left (-2 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)-2 x^2 \log (3)\right ) \log ^2(x)\right ) \log \left (e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}}+x\right )+\left (e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)+x^2 \log (3)\right ) \log ^2\left (e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}}+x\right )} \, dx=\frac {3}{-{\ln \left (x\right )}^2+\ln \left (x+{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-2}\,{\mathrm {e}}^{-\frac {x^2}{\ln \left (3\right )}}\right )+6} \] Input:
int(-(3*x*log(3) + exp((log(3)*(2*x - 2) - x^2)/log(3))*(6*x*log(3) - 6*x^ 2) - log(x)*(6*x*log(3) + 6*exp((log(3)*(2*x - 2) - x^2)/log(3))*log(3)))/ (log(x)^4*(x^2*log(3) + x*exp((log(3)*(2*x - 2) - x^2)/log(3))*log(3)) - l og(x)^2*(12*x^2*log(3) + 12*x*exp((log(3)*(2*x - 2) - x^2)/log(3))*log(3)) + log(x + exp((log(3)*(2*x - 2) - x^2)/log(3)))*(12*x^2*log(3) - log(x)^2 *(2*x^2*log(3) + 2*x*exp((log(3)*(2*x - 2) - x^2)/log(3))*log(3)) + 12*x*e xp((log(3)*(2*x - 2) - x^2)/log(3))*log(3)) + log(x + exp((log(3)*(2*x - 2 ) - x^2)/log(3)))^2*(x^2*log(3) + x*exp((log(3)*(2*x - 2) - x^2)/log(3))*l og(3)) + 36*x^2*log(3) + 36*x*exp((log(3)*(2*x - 2) - x^2)/log(3))*log(3)) ,x)
Output:
3/(log(x + exp(2*x)*exp(-2)*exp(-x^2/log(3))) - log(x)^2 + 6)
Time = 0.19 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.61 \[ \int \frac {-3 x \log (3)+e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} \left (6 x^2-6 x \log (3)\right )+\left (6 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} \log (3)+6 x \log (3)\right ) \log (x)}{36 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)+36 x^2 \log (3)+\left (-12 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)-12 x^2 \log (3)\right ) \log ^2(x)+\left (e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)+x^2 \log (3)\right ) \log ^4(x)+\left (12 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)+12 x^2 \log (3)+\left (-2 e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)-2 x^2 \log (3)\right ) \log ^2(x)\right ) \log \left (e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}}+x\right )+\left (e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}} x \log (3)+x^2 \log (3)\right ) \log ^2\left (e^{\frac {-x^2+(-2+2 x) \log (3)}{\log (3)}}+x\right )} \, dx=\frac {3}{\mathrm {log}\left (\frac {e^{\frac {x^{2}}{\mathrm {log}\left (3\right )}} e^{2} x +e^{2 x}}{e^{\frac {x^{2}}{\mathrm {log}\left (3\right )}} e^{2}}\right )-\mathrm {log}\left (x \right )^{2}+6} \] Input:
int(((6*log(3)*exp(((2*x-2)*log(3)-x^2)/log(3))+6*x*log(3))*log(x)+(-6*x*l og(3)+6*x^2)*exp(((2*x-2)*log(3)-x^2)/log(3))-3*x*log(3))/((x*log(3)*exp(( (2*x-2)*log(3)-x^2)/log(3))+x^2*log(3))*log(exp(((2*x-2)*log(3)-x^2)/log(3 ))+x)^2+((-2*x*log(3)*exp(((2*x-2)*log(3)-x^2)/log(3))-2*x^2*log(3))*log(x )^2+12*x*log(3)*exp(((2*x-2)*log(3)-x^2)/log(3))+12*x^2*log(3))*log(exp((( 2*x-2)*log(3)-x^2)/log(3))+x)+(x*log(3)*exp(((2*x-2)*log(3)-x^2)/log(3))+x ^2*log(3))*log(x)^4+(-12*x*log(3)*exp(((2*x-2)*log(3)-x^2)/log(3))-12*x^2* log(3))*log(x)^2+36*x*log(3)*exp(((2*x-2)*log(3)-x^2)/log(3))+36*x^2*log(3 )),x)
Output:
3/(log((e**(x**2/log(3))*e**2*x + e**(2*x))/(e**(x**2/log(3))*e**2)) - log (x)**2 + 6)